Challenge Problem 1 2016

If \({5}^{k^2}\times{25}^{2k}\times{625} = 25\times {\sqrt[4]{5}}\), which of the following is a possible value of \(k\)?

(A) \(-1\)

(B) \(-\Large\frac{1}{2}\)

(C) \(0\)

(D) \(\Large\frac{1}{2}\)

(E) \(1\)

First, notice that all the bases are powers of 5 – this includes the root which has exponent \(\frac{1}{4}\) (if you didn’t know that 625 is a power of 5, you need some more practice on Math FACTor 1). So, we can fiddle with the bases until we get something nicer: